Okay I just tested it on networked lobbies with someone and so far we found a few things.
1. It seems like the global healing between turns doesn't work.
2. Sometimes when attacking something it will reverse roles of defender / attacker. This has happened with both the person I was testing it with attacking me and me attacking an independent garrison.
3. I didn't receive a battle report when another player attacked me.
(edit)
It looks like its just anyone outside of the player 1 slot that doesnt get global healing between turns or resources either.
this also includes the first turn, youll have 0 herbs as druid for example even though you start with herb income and you'll stay at 0 herbs forever.